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\documentclass{article}
\usepackage{xeCJK}
\usepackage{amsmath}

\title{习题二T1(2,4),T2(1),T3(2,3),T4(2)}
\author{2020141460280张家帅}

\begin{document}
\maketitle
\section{习题二}
\subsection{T1}
\subsubsection{(2)}
设立如下谓词：

\begin{align}
       & P(x)\mbox{：}x\mbox{是直线}\notag           \\
       & Q(x,y)\mbox{：}x\mbox{和}y\mbox{平行}\notag \\
       & R(x,y)\mbox{：}x\mbox{和}y\mbox{相交}\notag
\end{align}

则有：
\[ (\forall x)(\forall y)((P(x)\wedge P(y))\rightarrow (Q(x,y)\leftrightarrow \neg R(x,y))) \]
\subsubsection{(4)}
设立如下谓词：

\begin{align}
       & P(x)\mbox{：}x\mbox{是正整数}\notag \\
       & Q(x)\mbox{：}x\mbox{是合数}\notag   \\
       & R(x)\mbox{：}x\mbox{是质数}\notag
\end{align}

则有：
\[ (\forall x)(P(x)\rightarrow(Q(x)\bigtriangledown R(x))) \]
\subsection{T2}
\subsubsection{(1)}
\[ (P(0)\wedge P(1)\wedge P(2))\wedge(Q(0)\vee Q(1)\vee Q(2)) \]
\subsection{T3}
\subsubsection{(2)}
第一个$\forall x$的辖域是$P(x)\wedge Q(x)$，$x$为约束变元，第二个$\forall x$的辖域为$P(x)$，$x$为约束变元，最后一个$Q(x)$为自由变元
\subsubsection{(3)}
$(\exists x)(\exists y)$的辖域为$P(x,y)\wedge Q(a)$，$x$和$y$是约束变元，$a$是自由变元，$\forall z$的辖域为$R(x,z)$，$x$是自由变元，$z$是约束变元
\subsection{T4}
\subsubsection{(2)}
\[ ((\forall x)[P(x)\rightarrow R(x)]\vee Q(y))\wedge((\exists x)R(x)\rightarrow(\exists z)S(y,z)) \]
\subsection{T5}
\subsubsection{(1)}
设立如下谓词：

\begin{align}
       & P(y)\mbox{：}y\mbox{是数}\notag  \\
       & Q(y)\mbox{：}y\mbox{等于}0\notag \\
       & x\mbox{为一个数}\notag
\end{align}

则有：
\[ (\forall y)(\forall z)((P(y)\wedge P(z))\rightarrow (Q(y)\vee Q(z))), \neg Q(x-1)\rightarrow \neg Q((x-1)(x+1)) \]

此公式为可满足公式
\subsubsection{(3)}
设立如下谓词：

\begin{align}
       & P(x)\mbox{：}x\mbox{是货物}\notag \\
       & Q(x)\mbox{：}x\mbox{是好的}\notag \\
       & R(x)\mbox{：}x\mbox{便宜}\notag   \\
       & a\mbox{：}小王买的衣服\notag
\end{align}

则有：
\[ (\forall x)((P(x)\wedge Q(x))\rightarrow \neg R(x)), \neg R(a)\rightarrow (P(a)\wedge Q(a)) \]

此公式为可满足公式
\subsection{T6}
\subsubsection{(1)}
公式$A$在$D$上为永真公式
\subsection{T7}
\subsubsection{(1)}
$x=0$时，公式为真，$x=-11$时，公式为假，所以此公式为可满足公式
\subsubsection{(2)}
$x=4$时，公式为真，此公式为永真公式
\subsection{T11}
\subsubsection{(1)}
\begin{align}
             & \neg((\forall x)P(x)\rightarrow(\exists y)P(y))\notag \\
      \Leftrightarrow{} & \neg(\neg(\forall x)P(x)\vee(\exists y)P(y))\notag    \\
      \Leftrightarrow{} & (\forall x)P(x)\wedge \neg(\exists y)P(y)\notag       \\
      \Leftrightarrow{} & (\forall x)P(x)\wedge (\exists y)\neg P(y)\notag      \\
      \Leftrightarrow{} & (\forall x)(\exists y)(P(x)\wedge\neg P(y))\notag     \\
      \Leftrightarrow{} & (\forall x)(P(x)\wedge\neg f(x))\notag
\end{align}
\subsubsection{(3)}
\begin{align}
             & (\forall x)(\forall y)[(\exists z)P(x,y,z)\wedge((\exists u)Q(x,u)\wedge(\exists v)Q(y,v))]\notag \\
      \Leftrightarrow{} & (\forall x)(\forall y)(\exists z)(\exists u)(\exists v)(P(x,y,z)\wedge Q(x,u)\wedge Q(y,v))\notag \\
      \Leftrightarrow{} & (\forall x)(\forall y)(P(x,y,f(x,y))\wedge Q(x,g(x,y))\wedge Q(y,h(x,y)))\notag
\end{align}
\end{document}